Why the Kirnberger Kernel Is So Small ∗ Don N. Page † Theoretical Physics Institute Department of Physics, University of Alberta Room 238 CEB, 11322 – 89 Avenue Edmonton, Alberta, Canada T6G 2G7 (2009 July 29) Abstract Defining the musical interval of the Kirnberger kernel, or Kirn-kern, to be one-twelfth the atom of Kirnberger, or the difference between a grad and a schisma, its natural logarithm, k = (161/12) ln 2 − 7 ln 3 − ln 5, is extremely small, k ≈ 0.000 000 739 401. Here an explanation of this coincidence is given −1 −1 by showing that k = (1/6)(11 tanh [(3/23)/11] − 21 tanh [(3/23)/21]) ≈ (255)/(3 · 72112233) ≈ 0.000 000 739 322. arXiv:0907.5249v1 [physics.pop-ph] 30 Jul 2009 ∗Alberta-Thy-14-09, arXiv:0907.5249 †Internet address: [email protected] 1 Introduction The historical accident that the most common musical scale today divides the octave into 12 semitones is no doubt due to the fact that 3 is close to 219/12, so that a justly tuned perfect fifth (pitch ratio 3:2 or 3/2) is very close to seven semitones of equal temperament (pitch ratio 27/12). The difference has been defined by Andreas Werck- meister (1645–1706) to be the grad [1], which (on a logarithmic scale) is one-twelfth of the Pythagorean or ditonic comma that is the difference between twelve perfect fifths and seven octaves (pitch ratio (3/2)12/27 =312/219 = 531441/524 288 ≈ 1.013 643). Having divided up the octave into twelve semitones so that a pitch ratio of three may be closely approximated by nineteen semitones, it is a rather fortuitous “accident” that a pitch ratio of five may also be rather closely approximated by an integer number of semitones, in this case twenty-eight. This is equivalent to the fact that a justly tuned major third (pitch ratio 5/4=1.25) is close to one-third of an octave or four semitones of equal temperament (pitch ratio 24/12 =21/3 ≈ 1.259 921), that is, the fact that the cube root of two is close to 5/4. Leaving aside the part of the coincidence that dividing up the octave into twelve semitones to get a good approximation in equal temperament to the justly tuned perfect fifth gives the cube root of two as an integer number of semitones, i.e., that the 3 denoting the cube root is an integer divisor of 12, the further coincidence that 21/3 ∼ 5/4 or 27 ∼ 53 is the basis of the fact that a kilobyte, 210 = 1 024 bytes, is close to one thousand bytes. That is, the ratio of a kilobyte to one thousand bytes is 27/53 =1.024, which in music is the pitch ratio called a diesis or minor diesis or enharmonic diesis [2]. However, there is an interesting further coincidence with these ratios. This is the fact that the difference between four semitones of equal temperament and a justly tuned major third (one third of a diesis on a logarithmic scale) is just very slightly more than an integer number, seven, times the difference between a justly tuned perfect fifth and seven semitones of equal temperament (a grad). The ratio between these two differences on a logarithmic scale is ln (27/3/5)/ ln (3/219/12) ≈ 7.000 655. That is, on a logarithmic scale, one-third of an enharmonic diesis is just very slightly more than seven times a grad (which is one-twelfth of a Pythagorean comma). A consequence of this coincidence is that 223/12/51/7 ≈ 3.000000316886 is very nearly three. One can then easily see that another consequence is that the twelfth root of two multiplied by the seventh root of five, which numerically to twelve-decimal- place accuracy is 1.333 333 192 495, is within about one part in ten million of being four-thirds. This coincidence was known to Johann Kirnberger (1721–1783), a student of Jo- hann Sebastian Bach, who developed [1] a rational intonation approximation to equal temperament by flattening the justly tuned perfect fifth by what is now called a schisma, the pitch ratio 385/215 = 32805/32768 = 1.001129150390625, rather than by the grad that is the pitch ratio 3/219/12 ≈ 1.001129890627526 that would be needed to get give exact equal temperament. (The musical sense of the word schisma was introduced by Alexander John Ellis (1814–1890), whom George Bernard Shaw acknowledged as the prototype of Professor Henry Higgins in his 1912 play Pygmalion [3].) Therefore, the Kirnberger fifth would be the 2 ratio 214/(375) = 16384/10935 ≈ 1.498308184728 rather than the equal tem- pered fifth of 27/12 ≈ 1.498 307 076 877. Twelve of these Kirnberger fifths ex- ceeds seven octaves, and therefore fails to close exactly, by the tiny interval of 21613−845−12 ≈ 1.000008872860, the atom of Kirnberger. 1 The size of the Kirnberger kernel Here I shall show why the atom of Kirnberger is so small, reducing the coincidence noted above to an coincidence involving only integers. For simplicity, I shall define the Kirnberger kernel, or Kirn-kern for short, to be (on a logarithmic scale) one- twelfth of the atom of Kirnberger. (The word kernel is used as the English cognate of the German word Kern, meaning “nucleus,” that I am here taking to be something smaller than an “atom.”) Now let me define lower-case letters for the natural logarithms of the various mu- sical intervals involved, and use the corresponding upper-case letters for the values of these quantities in the musical unit of a cent [4], which was also introduced by the Henry Higgins prototype Alexander Ellis and which corresponds to the musical interval of one-hundredth of a semitone, 21/1200 ≈ 1.000577790 and whose natural logarithm is c ≡ ln (2)/1200 ≈ 0.000577622650. (1) First, define the natural logarithm for an equal-temperament semitone as 1 h ≡ ln 2 ≈ 0.057762265047, (2) 12 with its value in cents being h H ≡ = 100. (3) c Second, define the natural logarithm of the grad as 19 g ≡ ln 3 − ln 2 ≈ 0.001129252782, (4) 12 with its value in cents being g G ≡ =1.955000865387. (5) c Third, define the natural logarithm of the Kirnberger kernel as 161 k ≡ ln 2 − 7 ln 3 − ln 5 ≈ 0.000000739402, (6) 12 with its value in cents being k K ≡ =0.001280077453. (7) c 3 I shall refer to these quantities as the semitone h, the grad g, and the Kirnberger kernel (or Kirn-kern) k, using the values of the natural logarithms instead of the pitch ratios themselves. The semitone h, grad g, and Kirn-kern k give a basis for writing the logarithm of any musical ratio formed from integer numbers of equal-temperament semitones, just perfect fifths, and just major thirds as a sum with integer coefficients of the h, g, and k. In particular, ln 2 = 12h, ln 3 = 19h + g, ln 5 = 28h − 7g − k, (8) so ln (2l/123m5n)=(l + 19m + 28n)h +(m − 7n)g − nk. (9) These basis numbers are also of greatly differing sizes, h ≈ 51.15g ≈ 78120k and g ≈ 1527k. It is not very surprising that the grad g is much smaller than the semitone, since the number of semitones in an octave was chosen to make it small, but it is rather surprising that the Kirn-kern is so much smaller even than the grad and that the ratio of the Kirn-kern to the grad is even smaller (by a factor of nearly thirty) than the ratio of the grad to the semitone. It is this smallness that I wish to explain in terms of a somewhat less surprising exact coincidence between integers. Define the natural logarithm of the enharmonic diesis (pitch ratio 128/125) to be 128 d ≡ ln = 7 ln 2 − 3 ln 5 = 21g +3k, (10) 125 and the natural logarithm of the syntonic comma (pitch ratio 81/80) to be 81 s ≡ ln = −4 ln 2 + 4 ln 3 − ln 5 = 11g + k. (11) 80 Since these musical intervals are considerably smaller than a semitone (being 41.06 and 21.51 cents respectively), it is not surprising that they can be written entirely in terms of the grad g and Kirn-kern k. One can readily solve these two equations to get the Kirn-kern k in terms of the enharmonic diesis d and syntonic comma s: 11 7 11 128 7 81 k = d − s = ln − ln . (12) 12 4 12 125 4 80 Next, we use the fact that the logarithm of a number just a bit larger than unity can be written in terms of the inverse hyperbolic tangent of a corresponding small number, which one can write as a power series in that small number. In particular, if we write 128/125 = (1+ x)/(1 − x) and 81/80= (1+ y)/(1 − y), then x = (128 − 125)/(128+125) = 3/253 = 3/(11 · 23) and y = (81 − 80)/(81+80) = 1/161 = 1/(7 · 23) = 3/(21 · 23), and one may write 1+ x −1 1 d = ln = 2 tanh x =2 x + x3 + ··· , (13) 1 − x 3 4 1+ y −1 1 s = ln = 2 tanh y =2 y + y3 + ··· . (14) 1 − y 3 Then one gets 1 −1 3 −1 3 k = 11 tanh − 21 tanh 6 11 · 23 21 · 23 1 3 3 1 3 3 3 3 = 11 − 21 + 11 − 21 + ··· 6 11 · 23 21 · 23 3 11 · 23 21 · 23 ! ! 255 160 ≈ = 3 · 72112233 216414429 ≈ 0.000000739322, (15) which is very tiny and is also within one part in nine thousand of the correct nu- merical value for the Kirnberger kernel, k ≈ 0.000000739402.